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<p>The paper aims at studying the long-time behavior of solution to the generalized complex mKdV equation. By analyzing the spectrum problem, a reconstruction formula of the solution associated with the Riemann-Hilbert problem is given. Using the nonlinear steepest descent method, various asymptotic formulas are obtained in the half-plane. More specifically, we get the leading asymptotic approximation of the solution in the slowly decaying oscillatory region, while the long-time asymptotics is also computed in the fast decaying region. Additionally, the Painlevé-type asymptotics is analyzed through the solution to a modified Painlevé II equation.</p>

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The Cauchy Problem for the Generalized Complex mKdV Equation: Long-Time and Painlevé Asymptotics

  • 期刊论文

  • 发表日期：

    2026年5月22日

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  • Wenhao Liu,
  • Yufeng Zhang

摘要

The paper aims at studying the long-time behavior of solution to the generalized complex mKdV equation. By analyzing the spectrum problem, a reconstruction formula of the solution associated with the Riemann-Hilbert problem is given. Using the nonlinear steepest descent method, various asymptotic formulas are obtained in the half-plane. More specifically, we get the leading asymptotic approximation of the solution in the slowly decaying oscillatory region, while the long-time asymptotics is also computed in the fast decaying region. Additionally, the Painlevé-type asymptotics is analyzed through the solution to a modified Painlevé II equation.